具有多重性的子空间覆盖

IF 0.9 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS
Anurag Bishnoi, Simona Boyadzhiyska, Shagnik Das, Tamás Mészáros
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引用次数: 5

摘要

摘要研究了余维$d$的仿射子空间的最小个数$f(n,k,d)$的确定问题,这些仿射子空间需要覆盖$\mathbb{F}_2^n\setminus \{\vec{0}\}$的所有点至少$k$次,而覆盖原点最多$k - 1$次。$k=1$是Jamison的经典结果,它是由browwer和Schrijver独立得出的$d = 1$。$f(n,1,1)$的值也来源于Alon和f redi关于仿射空间中任意域上有限网格覆盖的著名定理。在这里,我们在参数的不同范围内精确地确定这个函数的值。特别地,我们证明了对于$k\geq 2^{n-d-1}$我们有$f(n,k,d)=2^d k-\left\lfloor{\frac{k}{2^{n-d}}}\right\rfloor$,对于$n \gt 2^{2^d k-k-d+1}$我们有$f(n,k,d)=n + 2^d k -d-2$,并且得到了这两个范围之间的渐近结果。虽然在此方向上的先前工作主要采用多项式方法,但我们通过更直接的组合和概率论证证明了我们的结果,并且还利用了与编码理论的联系。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Subspace coverings with multiplicities
Abstract We study the problem of determining the minimum number $f(n,k,d)$ of affine subspaces of codimension $d$ that are required to cover all points of $\mathbb{F}_2^n\setminus \{\vec{0}\}$ at least $k$ times while covering the origin at most $k - 1$ times. The case $k=1$ is a classic result of Jamison, which was independently obtained by Brouwer and Schrijver for $d = 1$ . The value of $f(n,1,1)$ also follows from a well-known theorem of Alon and Füredi about coverings of finite grids in affine spaces over arbitrary fields. Here we determine the value of this function exactly in various ranges of the parameters. In particular, we prove that for $k\geq 2^{n-d-1}$ we have $f(n,k,d)=2^d k-\left\lfloor{\frac{k}{2^{n-d}}}\right\rfloor$ , while for $n \gt 2^{2^d k-k-d+1}$ we have $f(n,k,d)=n + 2^d k -d-2$ , and obtain asymptotic results between these two ranges. While previous work in this direction has primarily employed the polynomial method, we prove our results through more direct combinatorial and probabilistic arguments, and also exploit a connection to coding theory.
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来源期刊
Combinatorics, Probability & Computing
Combinatorics, Probability & Computing 数学-计算机:理论方法
CiteScore
2.40
自引率
11.10%
发文量
33
审稿时长
6-12 weeks
期刊介绍: Published bimonthly, Combinatorics, Probability & Computing is devoted to the three areas of combinatorics, probability theory and theoretical computer science. Topics covered include classical and algebraic graph theory, extremal set theory, matroid theory, probabilistic methods and random combinatorial structures; combinatorial probability and limit theorems for random combinatorial structures; the theory of algorithms (including complexity theory), randomised algorithms, probabilistic analysis of algorithms, computational learning theory and optimisation.
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